isprsarchives XL 5 W5 1 2015
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
CAMERA CONSTANT IN THE CASE OF TWO MEDIA PHOTOGRAMMETRY
P. Agrafiotis and A. Georgopoulos
National Technical University of Athens, School of Rural and Surveying Engineering, Lab. of Photogrammetry
Zografou Campus, Heroon Polytechniou 9, 15780, Zografou, Athens, Greece
(pagraf, drag)@central.ntua.gr
Commission V
KEY WORDS: Two media photogrammetry, Calibration, Camera constant, Refraction
ABSTRACT:
Refraction is the main cause of geometric distortions in the case of two media photogrammetry. However, this effect cannot be
compensated and corrected by a suitable camera calibration procedure (Georgopoulos and Agrafiotis, 2012). In addition, according
to the literature (Lavest et al. 2000), when the camera is underwater, the effective focal length is approximately equal to that in the
air multiplied by the refractive index of water. This ratio depends on the composition of the water (salinity, temperature, etc.) and
usually ranges from 1.10 to 1.34. It seems, that in two media photogrammetry, the 1.33 factor used for clean water in underwater
cases does not apply and the most probable relation of the effective camera constant to the one in air is depending of the percentages
of air and water within the total camera-to-object distance. This paper examines this relation in detail, verifies it and develops it
through the application of calibration methods using different test fields. In addition the current methodologies for underwater and
two-media calibration are mentioned and the problem of two-media calibration is described and analysed.
1. INTRODUCTION
Despite the growing interest of the photogrammetric community
for the transmission of a beam of light through media of
different density and hence different refractive index, refraction
remains the main cause of geometric distortions in the case of
multimedia photography. However, this effect cannot be
compensated and corrected by a suitable camera calibration
(Georgopoulos and Agrafiotis, 2012). This paper presents the
relation of the effective camera constant of two-media to the
one in air through the application of calibration methods using
different test fields. In addition the current methodologies for
underwater and two-media calibration are mentioned and the
problem of two-media calibration is described and analysed.
In the work of Lavest et al. (2000), it is proved that when the
camera is underwater, the effective focal length is
approximately equal to that in the air when multiplied by the
refractive index of water. This ratio depends on the composition
of the water (salinity, temperature, etc.) and usually ranges from
1.10 to 1.34. It seems, that in two media photogrammetry, the
1.33 factor used for clean water in underwater cases does not
apply and the most probable relation of the effective camera
constant to the one in air is depending of the percentages of air
and water in the total camera-to-object distance.
1980, Shan, 1994). According to the basic optical principles and
equations, one may consider that accurate through-water
photogrammetry is theoretically possible (Butler et al., 2002).
The geometry of two-media photogrammetry has been
addressed by Fryer and Kniest in 1985 and is presented in
Figure 1. Important is that for simplicity and as with other
studies, this research assumes that the water surface is planar.
Camera A
Camera B
B
A
Flying
Height
Normal
Normal
air
r
PB
PA
P'
P''
water
Apparent Depth
1.1 The Refraction Effect
Light rays passing through the flat interface of air and water
will be refracted according to Snell’s law according to which
the ratio of the sines of the angles of incidence and refraction is
equal to the ratio of phase velocities in the two media, or equal
to the reciprocal of the ratio of the indices of refraction:
Consequently, the value of this ratio is dependent on the optical
properties of the two media.
Two-media photogrammetry principles and geometry has been
widely addressed in the literature (Tewinkel 1963, Shmutter
and Monfigliolo, 1967, Masry, 1974, Karara, 1972, Slama,
i
Water
Height
Real Depth
P
Figure 1. The geometry of two-media Photogrammetry (Fryer
and Kniest, 1985)
In Figure 1 it is observed that light rays derived from an
underwater point P are refracted at the air/water interface
(which is assumed planar) at P A and P B and captured by Camera
A and Camera B respectively. If the systematic error resulting
from the refraction effect is ignored, then the two lines AP A and
This contribution has been peer-reviewed.
doi:10.5194/isprsarchives-XL-5-W5-1-2015
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
BP B do not intersect exactly on the normal, passing from the
underwater point P , but approximately at P'' , the apparent depth
of the point. Thus, without some form of correction, refraction
acts to produce an image of the surface which appears to lie at a
shallower depth than the real surface, and it is worthy of
attention that in each shot the Collinearity Condition is violated.
Light rays passing through the air/water interface are refracted
according to Snell’s law:
sin �
ℎ
=
=
sin � ℎ�
(1)
Where i is the angle of incidence of a light ray originating from
point P below the water surface, r is the angle of the refracted
ray above the water surface, h is the actual water depth, hA is the
apparent water depth and n is the refractive index, a value
depended on the optical properties of two media (Fryer and
Kniest, 1985).
1.2 Two Media Calibration: State of the Art
1.2.1 Underwater Camera Calibration: Underwater
camera calibration is also considered as two media calibration
since the camera is in a different medium from the calibration
test field. However, this case is differentiated from the through
water camera calibration by the amount of the percentages
intervening between the camera lens and the test field. For the
processing of underwater image data, two different approaches
are reported. One is based on the geometric interpretation for
light propagation through various media (e.g. air - housing
device - water) and the other on the application of suitable
corrections, in order to compensate for the refraction. Some
researchers use a pinhole camera for the estimation of the
refraction parameters (e.g.. Van der Zwaan et al., 2002; Pizarro
et al., 2004; Singh et al., 2005), while others calibrate the
cameras with the help of an object of known dimensions, which
is placed underwater in situ (Gracias and Santos-Victor, 2000;
Pessel et al., 2003; Höhle, 1971).
Self calibration is also applied for the camera-housing system,
where it is assumed that refraction effects are compensated by
the interior orientation parameters (Shortis and Harvey, 1998;
Gründig et al., 1999; Canciani et al., 2003; Harvey et al., 2003;
Drap et al., 2007; Shortis et al., 2007b). However, when
analyzing the correction of the image points, especially for close
ranges, the distance of the object points from the camera
severely affects the correction (Telem and Filin, 2010). Hence it
is concluded that a simple correction of the image points is not
adequate.
Finally, in Lavest et al. (2000) it is proved that it is possible to
infer the underwater calibration from the dry calibration by
multiplying the dry parameters by 1,333, the water refractive
index.
1.2.2 Through Water Calibration: It is characteristic of the
problem that most applications related to two-media
photogrammetry
and
specifically
through
water
photogrammetry do not overcome the refraction effects by a
camera calibration but try to correct them (Kotowski, 1988;
Westaway et al., 2001; Butler et al., 2002; Ferreira et al.,2006;
Murase et al., 2008; Mulsow, 2010; Georgopoulos and
Agrafiotis, 2012).
1.3 The problem of a suitable camera calibration
Unlike underwater photogrammetric procedures, where
according to the literature, the calibration is sufficient to correct
the effects of refraction, in two media cases, the sea surface
alteration due to waves (Fryer and Kniest, 1985, Okamoto,
1982), the solar reflections and the effects of refraction that
differ in each image, lead to unstable solutions. More
specifically, the amount of the refraction of a light beam is
affected by the amount of the water that covers the point of
origin and the angle of incidence of the beam in the air/water
interface. Therefore, the common calibration procedure of the
camera with a planar test field, fails to adequately describe the
effects of refraction in the photos in such cases.
A
A
z
z
y
rI
y
c
rI
rII
x
r
rI
x
image plane
r
c
rII
r
rII
rI
image plane
r
rII
air/water
interfaceII
H
air/water
interface I
i
H
i
Pa''
Pa'
i
i
Pb'
Pb''
hII
hI
Pa
Pb
Figure 2. The two-media Photogrammetry geometry for an
air/water interface I
Pa
Pb
Figure 3. The two-media Photogrammetry geometry for an
air/water interface II, in a higher level than I.
This contribution has been peer-reviewed.
doi:10.5194/isprsarchives-XL-5-W5-1-2015
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
In addition the shots from different angles and different
distances alter the amount of water that covers the points of the
test field, but also they significantly alter the angles of incidence
of the light rays, thus leading to unreliable results.
Hence, taking images with incident angles deviating largely
from the normal creates a sense of depth in the planar test filed
while substantially increasing the angle of incidence of the
optical beam. The result of this process is the overestimation of
the effects of refraction at certain positions through the
recording of an image of the test field that seems to lie at
shallower depth than the actual (apparent elevation) and
therefore it is imaged at a different scale. Consequently
calibration techniques do not succeed, as they fail to provide a
reliable result.
All necessary images were acquired with a Canon EOS MIII
full frame DSLR camera with a resolution of 21Mpixel
(5616x3744 pixels) and a 16 – 35 mm zoom lens, locked at
34mm, used for the experiment carried out in the simulation
tank and locked at 16mm, for the calibration in the field
conditions. While implementing the classical calibration
methodology using planar test fields, it was decided to rotate the
test fields and not move the camera around.
2. THEORETICAL APPROACH
In the photogrammetric and computer vision literature, the focal
length is defined as the distance between the camera sensor and
the optical center. It can be seen in Figure 2 and Figure 3 that
rays coming from two points, Pa and Pb, situated at the same
depth hI in Figure 2 and hII in Figure 3 and at the same distance
H from a Camera station A are refracted at the air/water planar
interface I and II and intersect the image plane at radial
distances rI and rII respectively.
In the described setup and in dry conditions, the rays coming
from these two points intersect the image plane at a radial
distance r, smaller than rI and rII. According to two-media
photogrammetry geometry and Snell's law, and taking into
account that the focal length is constant, it is derived that if hII
> hI, then rII > rI and always rII, rI > r . It is obvious that the
increase of the water depth implies a decrease of the solid angle
of view. This variation seems to be directly proportional to the
camera constant because the sensor size (and hence the image
size) is constant.
3. METHODOLOGY APPLIED
Relevant experiments were carried out in a simulation tank
filled with clean water and varying depth. This well controlled
experimental environment allowed the elimination of the errors
caused by waves, turbid water and sun glint and increased the
accuracy and reliability of the results. Finally, calibration results
from real two-media conditions are also presented.
3.1 Proving the misleading results of a camera calibration
in two media cases
For proving the misleading results of a camera calibration in
two media cases, three different calibration methods were
carried out in air and water in order to achieve objective results.
In addition, two different specially designed and made of
plexiglass, test fields were used. More specifically the following
camera calibration software suites or algorithms and test fields
(Figure 4) were used:
i. A specially developed calibration sheet of
Photomodeler® Scanner used for Photomodeler
Calibration Suite (www.photomodeler.com), with full
set of APs, and
ii. A specially developed calibration chessboard used for
calibrating the camera with the FAUCCAL (Douskos et
al., 2009.) and Camera Calibration Toolbox for Matlab®
(Bouguet, 2004).
(a)
(b)
Figure 4. Calibration boards for Photomodeler Scanner (a) and
FAUCCAL and Camera Calibration Toolbox (CCT) (b)
To this direction, images of the planar test fields with ±45° roll
angles were captured. This procedure is essential because in the
case of 2D calibration fields images must have been acquired
with significant tilts to produce strongly differing perspective
views of the plane. For the proper implementation of this
methodology, a special base was created in order to rotate the
test field in the specified different angles and not the camera. In
this way variations of the percentage of water and air in the
calibration images and of the incident angle are avoided. This
system primarily consists of a modified 3-way, pan/tilt
photographic tripod head in order to allow the already described
exact rotation of the mounted planar test fields. Also, the
percentages of air and water intervening between camera and
calibration fields were changed. It should be noted that camera
calibration was also applied only in dry conditions using all the
test fields in order to have a benchmark for the focal length and
the other parameters of the Interior Orientation of the camera.
3.2 Retrieving the Camera Constant in the case of Two
Media
Images of a specially graduated stable planar test field were
taken with increasing water percentages, each time by 5%,
maintaining the camera position and the test field constant in
order to recover the effective camera constant by measuring and
comparing the decreasing field of view. The images were also
checked for lens distortions. This final and more effective
procedure proves that in two media photogrammetry the
effective camera constant ratio to the one in air is proportional
to the percentages of air and water in the total camera-to-object
distance.
4. EXPERIMENTAL RESULTS
4.1 Camera calibration in two media cases
As already described, camera calibration in two media cases
using Photomodeler Calibration Suite, FAUCCAL and Camera
Calibration Toolbox for Matlab® was applied in order to prove
the resulting misleading results. The following presents these
results. Table 1 and Figure 5 present the results from the camera
calibration procedure with varying depths. It is obvious that the
solutions are not stable. Moreover, it could be observed that as
the percentage of the water intervening between the calibration
This contribution has been peer-reviewed.
doi:10.5194/isprsarchives-XL-5-W5-1-2015
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
field and the camera is increasing, the computed camera
constants differ from each other. This is the main problem
caused by the refraction effect in the calibration process.
Water
in %
0
15
30
45
Software
Photomodeler
33.92
46.27
46.29
46.07
FAUCCAL
34.05
46.22
46.32
43.94
CCT
34.07
46.43
45.8
45.15
Table 1. Camera constant results from experiments with 0, 15,
30, and 45% of intervening water between the camera and the
calibration field.
47
All the presented results prove the misleading results of a twomedia camera calibration using planar test fields. As already
stated, this is due to the fact that the amount of the refraction of
a light beam is affected by the amount of the water that covers
the point of origin and the angle of incidence of the beam in the
interface air/water. This varies with the rotations of the planar
test fields and leads to wrong results.
4.2 Camera Constant in the case of Two Media
As already mentioned, images of a specially graduated stable
planar test field (Figure 7) were taken with increasing water
percentages by 5%, while maintaining the camera - to - test field
distance constant. In that way, the effective camera constant can
be retrieved by measuring and comparing the decreasing field of
view and accordingly the scale.
46.5
46
45.5
Photomodeler
45
FAUCCAL
44.5
CCT
44
43.5
43
42.5
15%
30%
45%
Figure 5. The computed camera constant according to the water
percentages.
yp (mm)
(a)
RMS XYZ (mm)
xp (mm)
RMS xy (pixel)
Camera constant
c (mm)
Media
In Table 2, the results of a field calibration are presented and
were not as expected. It is noted that the camera constant in that
case is smaller than the one in air while it should be bigger.
However, in Table 1 it is noticed that the resulting camera
constants are greater than the one in air and the surprising fact is
that their ratio is more than the refractive index of the water
(1.33) in most of the cases. This could be observed better in
Figure 6.
air
16.519
17.901
12.045
0.212
0.061
air 15.483
17.743
11.683
6.262
2.310
water
Table 2. Camera Calibration results in air and air-water in field
conditions (Georgopoulos and Agrafiotis, 2012)
1.4
1.38
(b)
Figure 7. The graduated stable planar test field on 0% water (a)
and 25% water (b)
Translating the described theoretical approach into
mathematical terms, the camera constant coefficient is
expressed by:
1.36
1.34
Photomodeler
FAUCCAL
1.32
CCT
1.3
1.28
1.26
R15%
R30%
R45%
Figure 6. The ratio between the camera constant in two-media
and the one in air.
���� ×
���
+ ����
�
×
(2)
��� �
where, P air and P water are the percentages of air and water
respectively that intervene between camera and object, nair is the
refractive index of the air equals to 1 and n water the refractive
index of the water. The resulting coefficient values for
intervening water percentages up to 45% are presented in Figure
8 with the solid line. By the dashed line the experimental results
from our applied methodology are represented. In more detail,
This contribution has been peer-reviewed.
doi:10.5194/isprsarchives-XL-5-W5-1-2015
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
these results are derived by measuring the field of view of the
camera, while the water percentage was increased. By
measuring the field of view of an image containing water, the
ratio of the field of view in dry conditions by the measured field
of view was calculated.
1.2
Camera Constant coefficient
1.15
equals to 1 and n water the refractive index of the water. Moreover
it was proven in this paper that the camera calibration in twomedia photogrammetry leads to misleading results and thus the
scientific community should not overcome the refraction effects
by a camera calibration but they should try to correct those. The
aforementioned mathematical relationship bridges the gap
between the camera constant in air (dry) applications and the
camera constant in underwater applications, as described by
Lavest, Rives and Lapresté (Lavest et al., 2000).
ACKNOWLEDGEMENTS
1.1
1.05
Theory
1
Experiment
The research leading to these results has been supported by
European Union funds and National funds (GSRT) from Greece
and EU under the project 3D ORO: Efficient and Effective 3D
Computer Vision Tools for Improving the Performance of 3D
Digitalization funded under the cooperation framework.
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0.95
Bouguet, J. Y. (2004). Camera calibration toolbox for matlab.
0.9
0%
15%
20%
25%
40%
45%
Figure 8. Theoretical and Experimental coefficient comparison
regarding to the intervening water percentages
Theoretical
Experimental
Coefficient
Coefficient
0%
1
1
15%
1.0495
1.04
20%
1.066
1.068
25%
1.0825
1.09
40%
1.132
1.13
45%
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1.15
Table 3. resulting coefficients regarding to the intervening water
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1 �
=
�
(3)
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5. CONCLUSIONS
It is apparent, that in two media photogrammetry, the 1.33
factor used for clean water in underwater cases does not apply
and the relation of the effective camera constant to the one in air
is depending on the percentages of air and water within the total
camera-to-object distance. The effective camera constant to the
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an underwater stereovideo system for the monitoring of marine
fauna populations. International Archives Photogrammetry and
Remote Sensing, 32 (5), pp. 792-799.
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Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
CAMERA CONSTANT IN THE CASE OF TWO MEDIA PHOTOGRAMMETRY
P. Agrafiotis and A. Georgopoulos
National Technical University of Athens, School of Rural and Surveying Engineering, Lab. of Photogrammetry
Zografou Campus, Heroon Polytechniou 9, 15780, Zografou, Athens, Greece
(pagraf, drag)@central.ntua.gr
Commission V
KEY WORDS: Two media photogrammetry, Calibration, Camera constant, Refraction
ABSTRACT:
Refraction is the main cause of geometric distortions in the case of two media photogrammetry. However, this effect cannot be
compensated and corrected by a suitable camera calibration procedure (Georgopoulos and Agrafiotis, 2012). In addition, according
to the literature (Lavest et al. 2000), when the camera is underwater, the effective focal length is approximately equal to that in the
air multiplied by the refractive index of water. This ratio depends on the composition of the water (salinity, temperature, etc.) and
usually ranges from 1.10 to 1.34. It seems, that in two media photogrammetry, the 1.33 factor used for clean water in underwater
cases does not apply and the most probable relation of the effective camera constant to the one in air is depending of the percentages
of air and water within the total camera-to-object distance. This paper examines this relation in detail, verifies it and develops it
through the application of calibration methods using different test fields. In addition the current methodologies for underwater and
two-media calibration are mentioned and the problem of two-media calibration is described and analysed.
1. INTRODUCTION
Despite the growing interest of the photogrammetric community
for the transmission of a beam of light through media of
different density and hence different refractive index, refraction
remains the main cause of geometric distortions in the case of
multimedia photography. However, this effect cannot be
compensated and corrected by a suitable camera calibration
(Georgopoulos and Agrafiotis, 2012). This paper presents the
relation of the effective camera constant of two-media to the
one in air through the application of calibration methods using
different test fields. In addition the current methodologies for
underwater and two-media calibration are mentioned and the
problem of two-media calibration is described and analysed.
In the work of Lavest et al. (2000), it is proved that when the
camera is underwater, the effective focal length is
approximately equal to that in the air when multiplied by the
refractive index of water. This ratio depends on the composition
of the water (salinity, temperature, etc.) and usually ranges from
1.10 to 1.34. It seems, that in two media photogrammetry, the
1.33 factor used for clean water in underwater cases does not
apply and the most probable relation of the effective camera
constant to the one in air is depending of the percentages of air
and water in the total camera-to-object distance.
1980, Shan, 1994). According to the basic optical principles and
equations, one may consider that accurate through-water
photogrammetry is theoretically possible (Butler et al., 2002).
The geometry of two-media photogrammetry has been
addressed by Fryer and Kniest in 1985 and is presented in
Figure 1. Important is that for simplicity and as with other
studies, this research assumes that the water surface is planar.
Camera A
Camera B
B
A
Flying
Height
Normal
Normal
air
r
PB
PA
P'
P''
water
Apparent Depth
1.1 The Refraction Effect
Light rays passing through the flat interface of air and water
will be refracted according to Snell’s law according to which
the ratio of the sines of the angles of incidence and refraction is
equal to the ratio of phase velocities in the two media, or equal
to the reciprocal of the ratio of the indices of refraction:
Consequently, the value of this ratio is dependent on the optical
properties of the two media.
Two-media photogrammetry principles and geometry has been
widely addressed in the literature (Tewinkel 1963, Shmutter
and Monfigliolo, 1967, Masry, 1974, Karara, 1972, Slama,
i
Water
Height
Real Depth
P
Figure 1. The geometry of two-media Photogrammetry (Fryer
and Kniest, 1985)
In Figure 1 it is observed that light rays derived from an
underwater point P are refracted at the air/water interface
(which is assumed planar) at P A and P B and captured by Camera
A and Camera B respectively. If the systematic error resulting
from the refraction effect is ignored, then the two lines AP A and
This contribution has been peer-reviewed.
doi:10.5194/isprsarchives-XL-5-W5-1-2015
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
BP B do not intersect exactly on the normal, passing from the
underwater point P , but approximately at P'' , the apparent depth
of the point. Thus, without some form of correction, refraction
acts to produce an image of the surface which appears to lie at a
shallower depth than the real surface, and it is worthy of
attention that in each shot the Collinearity Condition is violated.
Light rays passing through the air/water interface are refracted
according to Snell’s law:
sin �
ℎ
=
=
sin � ℎ�
(1)
Where i is the angle of incidence of a light ray originating from
point P below the water surface, r is the angle of the refracted
ray above the water surface, h is the actual water depth, hA is the
apparent water depth and n is the refractive index, a value
depended on the optical properties of two media (Fryer and
Kniest, 1985).
1.2 Two Media Calibration: State of the Art
1.2.1 Underwater Camera Calibration: Underwater
camera calibration is also considered as two media calibration
since the camera is in a different medium from the calibration
test field. However, this case is differentiated from the through
water camera calibration by the amount of the percentages
intervening between the camera lens and the test field. For the
processing of underwater image data, two different approaches
are reported. One is based on the geometric interpretation for
light propagation through various media (e.g. air - housing
device - water) and the other on the application of suitable
corrections, in order to compensate for the refraction. Some
researchers use a pinhole camera for the estimation of the
refraction parameters (e.g.. Van der Zwaan et al., 2002; Pizarro
et al., 2004; Singh et al., 2005), while others calibrate the
cameras with the help of an object of known dimensions, which
is placed underwater in situ (Gracias and Santos-Victor, 2000;
Pessel et al., 2003; Höhle, 1971).
Self calibration is also applied for the camera-housing system,
where it is assumed that refraction effects are compensated by
the interior orientation parameters (Shortis and Harvey, 1998;
Gründig et al., 1999; Canciani et al., 2003; Harvey et al., 2003;
Drap et al., 2007; Shortis et al., 2007b). However, when
analyzing the correction of the image points, especially for close
ranges, the distance of the object points from the camera
severely affects the correction (Telem and Filin, 2010). Hence it
is concluded that a simple correction of the image points is not
adequate.
Finally, in Lavest et al. (2000) it is proved that it is possible to
infer the underwater calibration from the dry calibration by
multiplying the dry parameters by 1,333, the water refractive
index.
1.2.2 Through Water Calibration: It is characteristic of the
problem that most applications related to two-media
photogrammetry
and
specifically
through
water
photogrammetry do not overcome the refraction effects by a
camera calibration but try to correct them (Kotowski, 1988;
Westaway et al., 2001; Butler et al., 2002; Ferreira et al.,2006;
Murase et al., 2008; Mulsow, 2010; Georgopoulos and
Agrafiotis, 2012).
1.3 The problem of a suitable camera calibration
Unlike underwater photogrammetric procedures, where
according to the literature, the calibration is sufficient to correct
the effects of refraction, in two media cases, the sea surface
alteration due to waves (Fryer and Kniest, 1985, Okamoto,
1982), the solar reflections and the effects of refraction that
differ in each image, lead to unstable solutions. More
specifically, the amount of the refraction of a light beam is
affected by the amount of the water that covers the point of
origin and the angle of incidence of the beam in the air/water
interface. Therefore, the common calibration procedure of the
camera with a planar test field, fails to adequately describe the
effects of refraction in the photos in such cases.
A
A
z
z
y
rI
y
c
rI
rII
x
r
rI
x
image plane
r
c
rII
r
rII
rI
image plane
r
rII
air/water
interfaceII
H
air/water
interface I
i
H
i
Pa''
Pa'
i
i
Pb'
Pb''
hII
hI
Pa
Pb
Figure 2. The two-media Photogrammetry geometry for an
air/water interface I
Pa
Pb
Figure 3. The two-media Photogrammetry geometry for an
air/water interface II, in a higher level than I.
This contribution has been peer-reviewed.
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
In addition the shots from different angles and different
distances alter the amount of water that covers the points of the
test field, but also they significantly alter the angles of incidence
of the light rays, thus leading to unreliable results.
Hence, taking images with incident angles deviating largely
from the normal creates a sense of depth in the planar test filed
while substantially increasing the angle of incidence of the
optical beam. The result of this process is the overestimation of
the effects of refraction at certain positions through the
recording of an image of the test field that seems to lie at
shallower depth than the actual (apparent elevation) and
therefore it is imaged at a different scale. Consequently
calibration techniques do not succeed, as they fail to provide a
reliable result.
All necessary images were acquired with a Canon EOS MIII
full frame DSLR camera with a resolution of 21Mpixel
(5616x3744 pixels) and a 16 – 35 mm zoom lens, locked at
34mm, used for the experiment carried out in the simulation
tank and locked at 16mm, for the calibration in the field
conditions. While implementing the classical calibration
methodology using planar test fields, it was decided to rotate the
test fields and not move the camera around.
2. THEORETICAL APPROACH
In the photogrammetric and computer vision literature, the focal
length is defined as the distance between the camera sensor and
the optical center. It can be seen in Figure 2 and Figure 3 that
rays coming from two points, Pa and Pb, situated at the same
depth hI in Figure 2 and hII in Figure 3 and at the same distance
H from a Camera station A are refracted at the air/water planar
interface I and II and intersect the image plane at radial
distances rI and rII respectively.
In the described setup and in dry conditions, the rays coming
from these two points intersect the image plane at a radial
distance r, smaller than rI and rII. According to two-media
photogrammetry geometry and Snell's law, and taking into
account that the focal length is constant, it is derived that if hII
> hI, then rII > rI and always rII, rI > r . It is obvious that the
increase of the water depth implies a decrease of the solid angle
of view. This variation seems to be directly proportional to the
camera constant because the sensor size (and hence the image
size) is constant.
3. METHODOLOGY APPLIED
Relevant experiments were carried out in a simulation tank
filled with clean water and varying depth. This well controlled
experimental environment allowed the elimination of the errors
caused by waves, turbid water and sun glint and increased the
accuracy and reliability of the results. Finally, calibration results
from real two-media conditions are also presented.
3.1 Proving the misleading results of a camera calibration
in two media cases
For proving the misleading results of a camera calibration in
two media cases, three different calibration methods were
carried out in air and water in order to achieve objective results.
In addition, two different specially designed and made of
plexiglass, test fields were used. More specifically the following
camera calibration software suites or algorithms and test fields
(Figure 4) were used:
i. A specially developed calibration sheet of
Photomodeler® Scanner used for Photomodeler
Calibration Suite (www.photomodeler.com), with full
set of APs, and
ii. A specially developed calibration chessboard used for
calibrating the camera with the FAUCCAL (Douskos et
al., 2009.) and Camera Calibration Toolbox for Matlab®
(Bouguet, 2004).
(a)
(b)
Figure 4. Calibration boards for Photomodeler Scanner (a) and
FAUCCAL and Camera Calibration Toolbox (CCT) (b)
To this direction, images of the planar test fields with ±45° roll
angles were captured. This procedure is essential because in the
case of 2D calibration fields images must have been acquired
with significant tilts to produce strongly differing perspective
views of the plane. For the proper implementation of this
methodology, a special base was created in order to rotate the
test field in the specified different angles and not the camera. In
this way variations of the percentage of water and air in the
calibration images and of the incident angle are avoided. This
system primarily consists of a modified 3-way, pan/tilt
photographic tripod head in order to allow the already described
exact rotation of the mounted planar test fields. Also, the
percentages of air and water intervening between camera and
calibration fields were changed. It should be noted that camera
calibration was also applied only in dry conditions using all the
test fields in order to have a benchmark for the focal length and
the other parameters of the Interior Orientation of the camera.
3.2 Retrieving the Camera Constant in the case of Two
Media
Images of a specially graduated stable planar test field were
taken with increasing water percentages, each time by 5%,
maintaining the camera position and the test field constant in
order to recover the effective camera constant by measuring and
comparing the decreasing field of view. The images were also
checked for lens distortions. This final and more effective
procedure proves that in two media photogrammetry the
effective camera constant ratio to the one in air is proportional
to the percentages of air and water in the total camera-to-object
distance.
4. EXPERIMENTAL RESULTS
4.1 Camera calibration in two media cases
As already described, camera calibration in two media cases
using Photomodeler Calibration Suite, FAUCCAL and Camera
Calibration Toolbox for Matlab® was applied in order to prove
the resulting misleading results. The following presents these
results. Table 1 and Figure 5 present the results from the camera
calibration procedure with varying depths. It is obvious that the
solutions are not stable. Moreover, it could be observed that as
the percentage of the water intervening between the calibration
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
field and the camera is increasing, the computed camera
constants differ from each other. This is the main problem
caused by the refraction effect in the calibration process.
Water
in %
0
15
30
45
Software
Photomodeler
33.92
46.27
46.29
46.07
FAUCCAL
34.05
46.22
46.32
43.94
CCT
34.07
46.43
45.8
45.15
Table 1. Camera constant results from experiments with 0, 15,
30, and 45% of intervening water between the camera and the
calibration field.
47
All the presented results prove the misleading results of a twomedia camera calibration using planar test fields. As already
stated, this is due to the fact that the amount of the refraction of
a light beam is affected by the amount of the water that covers
the point of origin and the angle of incidence of the beam in the
interface air/water. This varies with the rotations of the planar
test fields and leads to wrong results.
4.2 Camera Constant in the case of Two Media
As already mentioned, images of a specially graduated stable
planar test field (Figure 7) were taken with increasing water
percentages by 5%, while maintaining the camera - to - test field
distance constant. In that way, the effective camera constant can
be retrieved by measuring and comparing the decreasing field of
view and accordingly the scale.
46.5
46
45.5
Photomodeler
45
FAUCCAL
44.5
CCT
44
43.5
43
42.5
15%
30%
45%
Figure 5. The computed camera constant according to the water
percentages.
yp (mm)
(a)
RMS XYZ (mm)
xp (mm)
RMS xy (pixel)
Camera constant
c (mm)
Media
In Table 2, the results of a field calibration are presented and
were not as expected. It is noted that the camera constant in that
case is smaller than the one in air while it should be bigger.
However, in Table 1 it is noticed that the resulting camera
constants are greater than the one in air and the surprising fact is
that their ratio is more than the refractive index of the water
(1.33) in most of the cases. This could be observed better in
Figure 6.
air
16.519
17.901
12.045
0.212
0.061
air 15.483
17.743
11.683
6.262
2.310
water
Table 2. Camera Calibration results in air and air-water in field
conditions (Georgopoulos and Agrafiotis, 2012)
1.4
1.38
(b)
Figure 7. The graduated stable planar test field on 0% water (a)
and 25% water (b)
Translating the described theoretical approach into
mathematical terms, the camera constant coefficient is
expressed by:
1.36
1.34
Photomodeler
FAUCCAL
1.32
CCT
1.3
1.28
1.26
R15%
R30%
R45%
Figure 6. The ratio between the camera constant in two-media
and the one in air.
���� ×
���
+ ����
�
×
(2)
��� �
where, P air and P water are the percentages of air and water
respectively that intervene between camera and object, nair is the
refractive index of the air equals to 1 and n water the refractive
index of the water. The resulting coefficient values for
intervening water percentages up to 45% are presented in Figure
8 with the solid line. By the dashed line the experimental results
from our applied methodology are represented. In more detail,
This contribution has been peer-reviewed.
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W5, 2015
Underwater 3D Recording and Modeling, 16–17 April 2015, Piano di Sorrento, Italy
these results are derived by measuring the field of view of the
camera, while the water percentage was increased. By
measuring the field of view of an image containing water, the
ratio of the field of view in dry conditions by the measured field
of view was calculated.
1.2
Camera Constant coefficient
1.15
equals to 1 and n water the refractive index of the water. Moreover
it was proven in this paper that the camera calibration in twomedia photogrammetry leads to misleading results and thus the
scientific community should not overcome the refraction effects
by a camera calibration but they should try to correct those. The
aforementioned mathematical relationship bridges the gap
between the camera constant in air (dry) applications and the
camera constant in underwater applications, as described by
Lavest, Rives and Lapresté (Lavest et al., 2000).
ACKNOWLEDGEMENTS
1.1
1.05
Theory
1
Experiment
The research leading to these results has been supported by
European Union funds and National funds (GSRT) from Greece
and EU under the project 3D ORO: Efficient and Effective 3D
Computer Vision Tools for Improving the Performance of 3D
Digitalization funded under the cooperation framework.
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0.95
Bouguet, J. Y. (2004). Camera calibration toolbox for matlab.
0.9
0%
15%
20%
25%
40%
45%
Figure 8. Theoretical and Experimental coefficient comparison
regarding to the intervening water percentages
Theoretical
Experimental
Coefficient
Coefficient
0%
1
1
15%
1.0495
1.04
20%
1.066
1.068
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1.0825
1.09
40%
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45%
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1.15
Table 3. resulting coefficients regarding to the intervening water
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�
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This contribution has been peer-reviewed.
doi:10.5194/isprsarchives-XL-5-W5-1-2015
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